Hauy Construction

Author

Eric W. Weisstein
September 27, 2017

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Polyhedra/HauyConstruction.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/HauyConstruction.html.

©2017 Wolfram Research, Inc. except for portions noted otherwise

Graphics

GraphicsRow[h1=Table[Show[HauyCuboctahedron[n],ViewPoint->{5,2,2},Boxed->False,Method->{"ShrinkWrap"->True}],{n,5}]]

GraphicsRow[h2=Table[Show[HauyOctahedron[n],ViewPoint->{5,2,2},Boxed->False,Method->{"ShrinkWrap"->True}],{n,1,9,2}]]

GraphicsRow[h3=Table[Show[HauyRhombicDodecahedron[n],ViewPoint->{5,2,2},Boxed->False,Method->{"ShrinkWrap"->True}],{n,1,17,4}]]

GraphicsRow[h4=Table[Show[HauySquarePyramid[n],ViewPoint->{5,2,2},Boxed->False,Method->{"ShrinkWrap"->True}],{n,5}]]

GraphicsGrid[{h1,h2,h3,h4},ImageSize->500]

Cube counts

Cuboctahedron

(seq=Table[Length[Solve[PolyhedronData["Cuboctahedron","RegionFunction"][x/n,y/n,z/n],{x,y,z},Integers]],{n,11}])//Timing

FindSequenceFunction[seq,n]

FindLinearRecurrence[seq]

FindGeneratingFunction[seq,x]

Octahedron

A001845

Table[HauyOctahedralNumber[k],{k,10}]

seq=Table[Length[HauyOctahedron[i][[1]]],{i,1,23,2}]

FindSequenceFunction[seq,n]//Factor

FindLinearRecurrence[seq]

LinearRecurrence[{4,-6,4,-1},{1,7,25,63},20]

FindGeneratingFunction[seq,x]

CoefficientList[Series[((((1+x))^3)/(((-1+x))^4)),{x,0,20}],x]

Limit[(((2n-1)(2n^2-2n+3)/3)/((n Sqrt[2])^3)),n->Infinity]

RhombicDodecahedron

A046142

seq=Table[HauyRhombicDodecahedralNumber[n],{n,10}]

HauyRhombicDodecahedron[5][[1]]

Length[%]

Table[Length[HauyRhombicDodecahedron[i][[1]]],{i,1,23,4}]

(wrong)

FindSequenceFunction[seq,n]//Factor

Table[(-1+2 n) (7-14 n+8 (n^2)),{n,20}]

FindLinearRecurrence[seq]

LinearRecurrence[{4,-6,4,-1},{1,33,185,553},20]

FindGeneratingFunction[seq,x]//Factor

CoefficientList[Series[((1+29 x+59 (x^2)+7 (x^3))/(((-1+x))^4)),{x,0,20}],x]

Limit[(((2n-1)(8n^2-14n+7))/((n Sqrt[2])^3)),n->Infinity]

SquarePyramid

A000447

seq=Length/@Table[HauySquarePyramid[n][[1]],{n,10}]

FindSequenceFunction[seq,n]

FindLinearRecurrence[seq]

seq=LinearRecurrence[{4,-6,4,-1},{1,10,35,84},{0,20}]

FindGeneratingFunction[seq,x]

CoefficientList[Series[((x (1+6 x+(x^2)))/(((-1+x))^4)),{x,0,20}],x]